The minimal ideal in multiplier algebras

Abstract

Let $\mathcal A$ be a simple, $\sigma$-unital, non-unital, non-elementary C*-algebra and let $I_{min}$ be the intersection of all the ideals of $\mathcal M(\mathcal A)$ that properly contain $\mathcal A$. $I_{min}$ coincides with the ideal defined by Lin (Simple C*-algebras with continuous scales and simple corona algebras. 112, (1991) Proc. Amer.Math. Soc) in terms of approximate units of $\mathcal A$ and $I_{min}/\mathcal A$ is purely infinite and simple. If $\mathcal A$ is separable, or if $\mathcal A$ has the (SP) property and its dimension semigroup $D(\mathcal A)$ of Murray-von Neumann equivalence classes of projections of $\mathcal A$ is order separable, or if $\mathcal A$ has strict comparison of positive elements by traces, then $\mathcal A\ne I_{min}$. If the tracial simplex $ \mathcal T(\mathcal A)$ is nonempty, let $ I_{con}$ be the closure of the linear span of the elements $A\in\mathcal M(\mathcal A)_+$ such that the evaluation map $\hat A(\tau)=\tau(A)$ is continuous. If $\mathcal A$ has strict comparison of positive element by traces then $I_{min}= I_{con}$. Furthermore, $I_{min}$ too has strict comparison of positive elements in the sense that if $A, B\in (I_{min})_+$, $B\not \in\mathcal A$ and $d_\tau(A)< d_\tau(B)$ for all $\tau\in \mathcal T(\mathcal A)$ for which $d_\tau(B)< \infty$, then $A\preceq B$. However if $\mathcal A$ does not have strict comparison of positive elements by traces then $I_{min}\ne I_{con}$ can occur: a counterexample is provided by Villadsen's AH algebras without slow dimension growth. If the dimension growth is flat, $ I_{con}$ is the largest proper ideal of $\mathcal M(\mathcal A)$.